Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-07T23:33:29.442Z Has data issue: false hasContentIssue false
  • English
  • Français

The ω-consistency of number theory via Herbrand's theorem

Published online by Cambridge University Press:  12 March 2014

W. D. Goldfarb  and
T. M. Scanlon
W. D. Goldfarb
Affiliation:
Society of Fellows, Harvard University, Cambridge, Massachusetts 02138
T. M. Scanlon
Affiliation:
Princeton University, Princeton, New Jersey 08540
Get access Rights & Permissions [Opens in a new window]

Extract

In this sequel to [7] the method of the consistency proof presented there is extended to provide a proof of the ω-consistency of the systems of number theory which were there shown consistent. This proof yields sharp bounds on the ordinal recursions required to establish the κ-consistency of these systems. The main technical innovation of this proof is the extension of what are essentially the methods of Ackermann [1] for handling finite sets of critical formulae of the first and second kinds to apply as well to sets of critical formulae in which the rank ordering is transfinite. The notation, definitions, and results of [7] will be presupposed throughout; we suggest the reader keep a copy of that paper at hand.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 4 , December 1974 , pp. 678 - 692
Copyright
Copyright © Association for Symbolic Logic 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ackermann, W., Zur Widerspruchsfreiheit der Zahlentheorie, Mathematische Annalen, vol. 117 (1940), pp. 162–194.Google Scholar
[2] Goldfarb, W. D., Ordinal bounds for κ-consistency, this Journal (to appear).Google Scholar
[3] Kreisel, G., On the interpretation of nonfinitist proofs. Part I, this Journal, vol. 16 (1951), pp. 241–267; Part II, this Journal, vol. 17 (1952), pp. 43–58.Google Scholar
[4] Kreisel, G., A survey of proof theory, this Journal, vol. 33 (1968), pp. 321–388.Google Scholar
[5] Kreisel, G. and Levy, A., Reflection principles and their use for establishing the complexity of axiom systems, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 97–142.Google Scholar
[6] Peter, R., Recursive functions, Academic Press, New York, 1967.Google Scholar
[7] Scanlon, T. M., The consistency of number theory via Herbrand's theorem, this Journal, vol. 38 (1973), pp. 29–58.Google Scholar
[8] Schütte, K., Beweistheorie, Springer, Berlin, 1960.Google Scholar
[9] Tait, W. W., Functionals defined by transfinite recursion, this Journal, vol. 30 (1965), pp. 155–174.Google Scholar